UNITED NATIONS, March 21 (Reuters) - In a major diplomatic blow for Syrian President Bashar al-Assad, Russia and China joined the U.N. Security Council on Wednesday in voicing support for U.N.-Arab League envoy Kofi Annan's bid to end violence that has brought Syria to the brink of civil war.

Western diplomats said the agreement on a statement voicing the "gravest concern at the deteriorating situation in Syria" should be a wake-up call for Assad, who has counted on support from his ally Russia to fend off international criticism of his year-long attempt to crush anti-government protests.

The statement threatens Syria with "further steps" if it fails to comply with Annan's six-point peace proposal, which calls for a cease-fire, political dialogue between government and opposition, and full access for aid agencies.

The U.S.-European push for a council statement backing Annan's mission came after Russia and China twice vetoed resolutions condemning Assad's assault on demonstrators, which the United Nations says has killed well over 8,000 civilians.

Unlike resolutions, which are legally binding and need nine votes in favor and no vetoes from the five permanent council members to pass, statements are generally non-binding but require unanimous support from the council.

Although the statement does not explicitly back an Arab League plan calling for Assad to step aside, it does include Annan's call for a political process that echoes that plan.

It voices "full support for the efforts of (Annan) to bring an immediate end to all violence and human rights violations, secure humanitarian access, and facilitate a Syrian-led political transition to a democratic, plural political system."

The statement also demands that the Syrian government stop fighting first, something Annan and the West have called for.

"The Syrian government should immediately cease troop movements towards, and end the use of heavy weapons in, population centers, and begin pullback of military concentrations in and around population centers," it said.

Once the government forces stop fighting, Syrian authorities "should work with (Annan) to bring about a sustained cessation of armed violence in all its forms by all parties."

"PRAGMATIC LOOK"

Western powers diluted the 15-nation council's "presidential statement" on Tuesday in an effort to secure the support of Russia, which had disliked language in an earlier version that it said sounded like an ultimatum for Syria, diplomats said.

Speaking at a news conference in Berlin shortly before the statement was agreed to, Russian Foreign Minister Sergei Lavrov made clear that Moscow fully supported the amended text.

"The council text reflects the reality in Syria and supports Annan's aims," Lavrov said after talks in Berlin with his German and Polish counterparts. "We support it fully."

"The most important thing is that there are no ultimatums ... and no suggestions as to who carries more blame," he said.

In New York, Russian U.N. Ambassador Vitaly Churkin said the council had "finally chosen to take a pragmatic look at Syria."

British U.N. Ambassador Mark Lyall Grant said the statement sends a "strong and united message to the government and all other actors in Syria that they need to respond ... immediately" to Annan's peace proposals.

German Ambassador Peter Wittig told reporters that he welcomed the "newly found unity of the council."

The council also approved informal remarks to the press that Russia had asked the council to issue condemning bomb attacks last weekend in the cities of Damascus and Aleppo.

The presidential statement is separate from a U.S.-drafted resolution calling on Syria to allow access to humanitarian aid workers in the country.

The last time the council passed a presidential statement on Syria was August 2011, although council members reached a rare unanimous agreement on informal remarks to the press on March 1 to rebuke Damascus for not allowing U.N. humanitarian aid chief Valerie Amos into the country.

Shortly after the council approved those remarks to the press, Amos was allowed to visit Damascus.